Solve the equation. $\dfrac{dy}{dx}=\dfrac{\cos(x)}{\sin(y)}$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=\arcsin(-\sin(x))+C$ (Choice B) B $y=\arcsin(-\sin(x)+C)$ (Choice C) C $y=\arccos(-\sin(x))+C$ (Choice D) D $y=\arccos(-\sin(x)+C)$
Explanation: We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=\dfrac{\cos(x)}{\sin(y)} \\\\ \sin(y)\,dy&=\cos(x)\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} \sin(y)\,dy&=\cos(x)\,dx \\\\ \int \sin(y)\,dy&=\int \cos(x)\,dx \\\\ -\cos(y)&=\sin(x)+C_1 \\\\ \cos(y)&=-\sin(x)+C \\\\ \arccos(\cos(y))&=\arccos(-\sin(x)+C) \\\\ y&=\arccos(-\sin(x)+C) \end{aligned}$ [Where did we get C?] Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=\arccos(-\sin(x)+C)$